Kakuro Combinations: Sum Tables and How to Use Them

What a sum table can and cannot tell you

Every lookup needs two inputs: the clue total and the number of cells in its run. A total of 6 over two cells allows {1,5} or {2,4}; the same total over three cells allows only {1,2,3}. Looking up the total without counting cells produces incorrect candidates.

The entries below are unordered sets. The combination {2,5} represents either 2 followed by 5 or 5 followed by 2. A table deliberately avoids choosing between these arrangements. Each cell's crossing run supplies the positional restrictions that a sum alone cannot provide.

Repeated digits are excluded, including arithmetic possibilities such as 3 + 3 = 6. Digits outside 1–9 are also excluded. A pair totaling 11 cannot include {1,10}. When using combinations for a partially filled run, remove any set containing a digit already entered elsewhere in that run.

Complete two-cell Kakuro combination table

Two-cell sums range from 3 to 17. For each total, the table lists every unordered pair of different digits from 1 to 9. Braces group the digits in a combination; a semicolon separates alternative combinations. A row with one combination fixes the digit set, but usually leaves its order unresolved.

Work a table lookup into a deduction

Take a two-cell across run totaling 11, with cells A and B. The table supplies four possibilities: {2,9}, {3,8}, {4,7}, and {5,6}. Suppose crossing clues reduce A to {2,4} and B to {3,7,9}. Evaluate each complete pair rather than mixing digits from unrelated options.

The pair {2,9} works as A = 2 and B = 9. The pair {4,7} works as A = 4 and B = 7. The other pairs fail because A cannot take either of their digits. Your surviving assignments are therefore (2,9) and (4,7); B's candidate 3 can be removed.

If a later down-run deduction removes 9 from B, only (4,7) remains. A becomes 4 and B becomes 7. Notice the two stages: the first lookup reduced uncertainty without solving the run, and the later crossing information selected a complete assignment.

Useful three-cell combinations

Three-cell sums range from 6 to 24. The endpoints are forced sets: 6 requires {1,2,3}, while 24 requires {7,8,9}. The nearby totals are also restrictive. The small table gives all combinations for the listed totals, rather than a complete table for every possible three-cell clue.

Use these entries exactly as you use two-cell pairs. If a three-cell run totaling 8 has a confirmed 5, the set must be {1,2,5}, so the other cells are 1 and 2 in some order. If a crossing instead forbids 5 throughout the run, the surviving set is {1,3,4}.

Generate longer combinations systematically

You do not need a giant table to handle a longer run. Write digits in increasing order when generating sets. For a three-cell total, choose the smallest digit first, then a larger second digit; calculate the third by subtraction. Keep the set only if the third is larger again and at most 9.

For example, a three-cell total of 10 gives {1,2,7}, {1,3,6}, {1,4,5}, and {2,3,5}. Increasing order prevents duplicates such as listing {7,1,2} as a new combination. It also rejects repeated digits automatically when the calculated final digit is not strictly larger than the previous one.

For any run length, check the smallest and largest distinct-digit totals before enumerating. Four cells need at least 1 + 2 + 3 + 4 = 10 and at most 6 + 7 + 8 + 9 = 30. A four-cell clue outside that range indicates a misread clue or a mistake.

Very long runs have a useful complement. The digits 1–9 total 45, so an eight-cell run totaling 37 omits exactly one digit: 45 − 37 = 8. Its set is therefore {1,2,3,4,5,6,7,9}. You still need crossing clues to place those eight digits.

Use combinations without overfilling your notes

Record full sets beside difficult runs and smaller candidate lists inside individual cells. If you copy every digit from every set into every cell without checking crossings, your notes become noisy and can hide the useful restrictions. Update the run and its cell notes together after a confirmed entry.

To check a completed run, add its actual digits and verify uniqueness. Matching one entry in a combination table is only part of the check: each digit must also satisfy its crossing run. A locally valid set can be arranged in an order that breaks the rest of the grid.

1. Count the white cells before choosing a table row.
2. Keep all listed alternatives until a specific constraint excludes them.
3. Filter out digits already used in the same run.
4. Test whether each set can be assigned to the actual cell candidates.

Frequently asked questions